@acer 
Are you talking about G1[iii]? I couldn't think of anything else. After all, each frame produces a new equation, and it needs to be displayed on the graph. And even then, the animation is shown after 2 frames due to resource limitations.

@Rouben Rostamian  No, it's much simpler. This is computational geometry. A set of points is rigidly attached to a curved (small) torus, and a trajectory is laid out on the large torus, along which the small torus supposedly rolls. At each point of the trajectory, calculations are made so that the small torus imitates rolling without slipping. This is done using Euler angles: we recalculate the coordinates of the attached points, and then, based on this, we transform the equation of the small torus and draw another graph of motion.
I'll never be able to handle physics equations.

@Rouben Rostamian  
I thought  if we move along the graph line itself, then the proximity between the roots has a weaker effect on the situation with their possible omission. That is, we are talking about the values ​​y(s) and x(s), where s is a natural parameter (arc length). And then we can, for example, use a cycle "to count" harmonics based on their average length. And each time start counting from the last point on the curve. Then we should equally successfully pass "empty" ones, where y does not change sign, and places with a cluster of roots. In this case, most likely, it is necessary to additionally clarify each last point on the line at each step of the supposed cycle. 

(edited)
No, it looks like it won't be more reliable than your proposal. To move along the curve, an unacceptable number of significant digits would be required. "My" method is already very poorly controlled beyond 10^5

@Rouben Rostamian  

I don't need all the solutions of the cluster. I want to make sure that at a significant distance, for example, at a distance of 10^5 - 10^6 from the origin, not a single root out of about 50 roots in a row is guaranteed to be skipped. I'm just comparing with the capabilities of that very method of continuation by parameter. Here, I recently picked up an example. I'm already embarrassed that I attracted so much attention to myself, but it is difficult for me to satisfy my curiosity on my own due to my knowledge of the package.

@Rouben Rostamian  
You understood everything correctly. The only thing is, I wanted to see not the closest, in your words, cluster, but the most distant one, where it is still possible to find solutions in a row.
(Well, I use this very translator.☺)

@mmcdara  
Everything is correct, only if the following condition is met: at least 50 roots in a row (not a single missing root) in its environment.

@mmcdara  
Would I really suggest this example if the solutions were to be sought at such a "significant" distance from 0? ☺ It is need to find not the first 52 solutions, but the last ones that you can. 
By the way, if you decide to try again, please consider the capabilities of Maple 17.

Confirmation of problems with the Schatz mechanism. 

@C_R 
I don't know, such expressions make me doubt the reality of the model. The only thing, it made me remember the way of implementing Draghilev's method (I already mentioned  here), when we do not calculate the minors of the Jacobian matrix in symbolic form. I have some Pascal (Delphi) texts that I translated from FORTRAN at one time. One numerical method for solving ODEs (the Hamming method) is implemented there. The point, of course, is not in the method of solving ODE. But there, precisely because of the limited symbolic capabilities of more lower-level languages, only the Jacobi matrix was used (calculated manually). All left-hand sides of the ODE are obtained numerically: at each step, a system of linear equations is solved.
The calculations are quite fast. And, most importantly, we are much less dependent on the dimensionality of the original system of equations and on the cumbersome form of partial derivatives, because we do not perform symbolic transformations.
It is not yet clear to me how to adequately translate this approach to Maple.. So far, the only thing in my head is to rewrite the text from Pascal to Maple. I'm thinking about it.

@C_R 
I sometimes use the op and lhs functions. I just can't execute the program and apply them, for example, to eqm2. My computer freezes right at the start.
So far, from what you've said, I can understand that we have 3 equations and 4 variables, and very cumbersome expressions, right? Of course, the Method is quite applicable for such dimensions, but we need an appropriate computer at least to see the equations themselves. Apparently, your computer is suitable for this. Then we can use the Method, avoiding working with large formulas, stopping at the matrix of partial derivatives, but without calculating the determinant in symbolic form.

@C_R 
Just saw your request. I can't understand from the text what form these equations are.
Right, these equations eqm, eqm2, eqTPO? I don't see their content. Can you show me their contents separately?

@C_R 
Yes, the second question touches upon the strongest side of the Method. For example, in that very Schatz mechanism, one equation was discarded from the system of equations describing geometric connections, and we obtained a motion with small deformations. To generalize, we can find the infinite subset of solutions. After all, the set of solutions can break down into disconnected subsets (branches), and we need to find each branch. But it also happens that the entire set of solutions is connected, and in this case we solve the system at once, that is, in this case there is only one "pink" line. But we always need one starting point on the pink line, because this pink line is the solution to the Cauchy problem.
As for the first question (finding point solutions of the NxN system), the Method cannot guarantee a complete solution head-on, there are just successful examples. It is necessary to try to work with the system of equations in parts, as if working with a part of its equations. But even applying the Method head-on to NxN systems has undeniable advantages, the only thing, as it seems to me, is that it may be inferior to modern optimization methods for individual solutions. Therefore, optimization methods or Gröbner bases for polynomial systems of small dimension are very convenient for finding the starting point for pink lines.

@dharr  Yes, I also think that the successful solution of the original transcendental system  using polynomial theory is a fortunate coincidence of many circumstances.
Thanks again.

@dharr With your permission, I moved your comment to the answers, and +.
Thank you, there is something to learn about working with Maple.

@vv Yes, I used the Groebner package and figured out plex, but these were very simple and small examples. My PC got to your nops(G) in a reasonable time, but I'm a bit scared to try to go further.
Agree that it is possible to bring the work to the solution of a polynomial system only if there is a very serious interest. You have satisfied my curiosity completely.
If you are interested, at one time (back in the era of large computers) a group of several people was created for this task. True, I don’t know anything about the results.